Quadratic Rational Maps Research Articles

Moduli Spaces of Quadratic Maps: Arithmetic and Geometry

Abstract We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$th Gleason polynomial $G_{n}\in{\mathbb{Q}}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $\operatorname{Per}_{n}(0)$ is the $1$-dimensional moduli space of quadratic rational maps on ${\mathbb{P}}^{1}$ with an $n$-periodic critical point. We show that if $G_{n}$ is irreducible over ${\mathbb{Q}}$, then $\operatorname{Per}_{n}(0)$ is irreducible over ${\mathbb{C}}$. To do this, we exhibit a ${\mathbb{Q}}$-rational smooth point on a projective completion of $\operatorname{Per}_{n}(0)$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $n$, $\operatorname{Per}_{n}(0)$ itself has no ${\mathbb{Q}}$-rational points.

Dynatomic Galois groups for a family of quadratic rational maps

For every nonconstant rational function [Formula: see text], the Galois groups of the dynatomic polynomials of [Formula: see text] encode various properties of [Formula: see text] are of interest in the subject of arithmetic dynamics. We study here the structure of these Galois groups as [Formula: see text] varies in a particular one-parameter family of maps, namely, the quadratic rational maps having a critical point of period 2. In particular, we provide explicit descriptions of the third and fourth dynatomic Galois groups for maps in this family.

Birational Quadratic Planar Maps with Generalized Complex Rational Representations

Complex rational maps have been used to construct birational quadratic maps based on two special syzygies of degree one. Similar to complex rational curves, rational curves over generalized complex numbers have also been constructed by substituting the imaginary unit with a new independent quantity. We first establish the relationship between degree one, generalized, complex rational Bézier curves and quadratic rational Bézier curves. Then we provide conditions to determine when a quadratic rational planar map has a generalized complex rational representation. Thus, a rational quadratic planar map can be made birational by suitably choosing the middle Bézier control points and their corresponding weights. In contrast to the edges of complex rational maps of degree one, which are circular arcs, the edges of the planar maps can be generalized to hyperbolic and parabolic arcs by invoking the hyperbolic and parabolic numbers.

Equidistribution for matings of quadratic maps with the modular group

AbstractWe study the asymptotic behavior of the family of holomorphic correspondences $\lbrace \mathcal {F}_a\rbrace _{a\in \mathcal {K}}$ , given by $$ \begin{align*}\bigg(\frac{az+1}{z+1}\bigg)^2+\bigg(\frac{az+1}{z+1}\bigg)\bigg(\frac{aw-1}{w-1}\bigg)+\bigg(\frac{aw-1}{w-1}\bigg)^2=3.\end{align*} $$ It was proven by Bullet and Lomonaco [Mating quadratic maps with the modular group II. Invent. Math.220(1) (2020), 185–210] that $\mathcal {F}_a$ is a mating between the modular group $\operatorname {PSL}_2(\mathbb {Z})$ and a quadratic rational map. We show for every $a\in \mathcal {K}$ , the iterated images and preimages under $\mathcal {F}_a$ of non-exceptional points equidistribute, in spite of the fact that $\mathcal {F}_a$ is weakly modular in the sense of Dinh, Kaufmann, and Wu [Dynamics of holomorphic correspondences on Riemann surfaces. Int. J. Math.31(05) (2020), 2050036], but it is not modular. Furthermore, we prove that periodic points equidistribute as well.

Quasi-adelic measures and equidistribution on

AbstractBaker and Rumely, Favre, Rivera, and Letelier, and Chambert-Loir proved an important arithmetic equidistribution theorem for points of small height associated to an adelic measure. To broaden the scope in which arithmetic equidistribution may be employed, we generalize the notion of an adelic measure to that of a quasi-adelic measure and show that arithmetic equidistribution holds for quasi-adelic measures as well. We exhibit examples of non-adelic, quasi-adelic measures arising from the dynamics of quadratic rational maps. In fact, we show that the measures that arise in applications of arithmetic equidistribution theorems are typically not adelic. Finally, we motivate our definition of a quasi-adelic measure by relating it to a seemingly different problem in arithmetic dynamics arising from results of Call, Tate, and Silverman in the study of abelian varieties.

Quadratic rational maps with integer multipliers

In this article, we prove that every quadratic rational map whose multipliers all lie in the ring of integers of a given imaginary quadratic field is a power map, a Chebyshev map or a Lattès map. In particular, this provides some evidence in support of a conjecture by Milnor concerning rational maps that have an integer multiplier at each cycle.

Quadratic Rational Maps with a 2-Cycle of Siegel Disks

For the family of quadratic rational functions having a $2$-cycle of bounded type Siegel disks, we prove that each of the boundaries of these Siegel disks contains at most one critical point. In the parameter plane, we prove that the locus for which the boundaries of the $2$-cycle of Siegel disks contain two critical points is a Jordan curve.

Iterated nth order nonlinear quantum dynamics with mixed initial states

Nonlinear quantum transformations play an important role in quantum state distillation, aiding quantum communication schemes. The simplest of these involve quadratic rational maps for qubits, higher order versions have not been explored in detail. We consider such iterated nonlinear protocols for qubits, involving a generalized CNOT, a measurement, and a single-qubit Hadamard gate processing n inputs in identical quantum states. To characterize the global properties of these protocols, we determine the fixed points and relevant fixed cycles. We show that a repelling mixed fixed point exists for all orders, marking a phase transition related to the disappearance of the fractalness of borders of different basins of attraction.

Monotonicity of entropy for unimodal real quadratic rational maps

<p style='text-indent:20px;'>We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients and real critical points. This confirms a conjecture made in [<xref ref-type="bibr" rid="b8">8</xref>].</p>

Slices of parameter space for meromorphic maps with two asymptotic values

AbstractThis paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work by Goldberg and Keen [The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift.Invent. Math.101(2) (1990), 335–372] on degree two rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further is that, despite the presence of the essential singularity, our new functions exhibit a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted.

Monotonicity of entropy for real quadratic rational maps

The monotonicity of entropy is investigated for real quadratic rational maps on the real circle based on the natural partition of the corresponding moduli space into its monotonic, covering, unimodal and bimodal regions. Utilizing the theory of polynomial-like mappings, we prove that the level sets of the real entropy function are connected in the (−+−)-bimodal region and a portion of the unimodal region in . Based on the numerical evidence, we conjecture that the monotonicity holds throughout the unimodal region, but we conjecture that it fails in the region of (+−+)-bimodal maps.

Constructing quadratic birational maps via their complex rational representation

We present a method to construct quadratic birational planar maps, which is based on the observation that any rational planar map with complex rational representation possesses two special syzygies. After establishing the relation between degree one complex rational Bézier curves and quadratic rational Bézier curves, we derive conditions to determine when a quadratic rational planar map has a complex rational representation. Hence, we can make a quadratic planar map be birational by suitably choosing the middle Bézier control points and their corresponding weights. In addition, this paper explores an interesting geometric property of the isoparametric curves of quadratic birational planar maps with complex rational representation, i.e., all isoparametric curves meet at a common point.

Iterated towers of number fields by a quadratic map defined over the Gaussian rationals

An iterated tower of number fields is constructed by adding preimages of a base point by iterations of a rational map. A certain basic quadratic rational map defined over the Gaussian number field yields such a tower of which any two steps are relative bicyclic biquadratic extensions. Regarding such towers as analogues of $\mathbf{Z}_{2}$-extensions, we examine the parity of 2-ideal class numbers along the towers with some examples.

Regular polynomial automorphisms in the space of planar quadratic rational maps

In this paper, we describe the semistable quotient of the set of regular polynomial automorphisms $${\mathcal {H}}_2^2$$ in the semistable locus of the moduli space of quadratic rational maps, using the portrait moduli space of rational maps with a fixed point. We also provide the parametrization of $${\mathcal {H}}_2^2$$ using two invariants, $$\det f$$ and $${\text {tr}}\,f$$ .

Julia Sets of Cubic Rational Maps with Escaping Critical Points

It is known that the Julia set of a quadratic rational map is either connected or a Cantor set. In this paper, we explore this dichotomy for the maps in a type of three-dimensional space of cubic rational maps. We show that for a cubic rational map f, if f has an attracting fixed point p and all critical points are attracted to p under the iteration of f, then the Julia set J(f) is either a Cantor set or a connected set (and locally connected) with one possible exception; we also give a necessary and sufficient condition for J(f) to be a Sierpinski curve when it is connected.

On the non-monotonicity of entropy for a class of real quadratic rational maps

We prove that the entropy function on the moduli space of real quadratic rational maps is not monotonic by exhibiting a continuum of disconnected level sets. This entropy behavior is in stark contrast with the case of polynomial maps, and establishes a conjecture on the failure of monotonicity for bimodal real quadratic rational maps of shape $(+-+)$ which was posed in arXiv:1901.03458 based on experimental evidence.

On the Multipliers at Fixed Points of Quadratic Self-Maps of the Projective Plane with an Invariant Line

This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of $\mathbb{P}^2$ and quadratic homogeneous vector fields on $\mathbb{C}^3$, the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on $\mathbb{C}^3$ having exclusively single-valued solutions.

Quadratic Thurston maps with few postcritical points

We use the theory of self-similar groups to enumerate all combinatorial classes of non-Euclidean quadratic Thurston maps with fewer than five postcritical points. The enumeration relies on our computation that the corresponding maps on moduli space can be realized by quadratic rational maps with fewer than four postcritical points.

Critical orbits of polynomials with a periodic point of specified multiplier

Answering a question posed by Adam Epstein, we show that the collection of conjugacy classes of polynomials admitting a parabolic fixed point and at most one infinite critical orbit is a set of bounded height in the relevant moduli space. We also apply the methods over function fields to draw conclusions about algebraically parametrized families, and prove an analogous result for quadratic rational maps.

A special class of infinite measure-preserving quadratic rational maps

ABSTRACTWe show the existence of a set of positive measure in parameter space, each element corresponding to a rational map of degree , for which there exists an invariant infinite σ-finite measure equivalent to a conformal measure. These maps are conservative and exact with respect to the invariant measure, and are perturbations of generalized Boole maps and inner functions. For quadratic Boole maps of with real coefficients we show that the Krengel entropy provides a complete invariant for c-isomorphism classes of quadratic maps of this type.